A079289 -id:A079289 - cp's OEIS frontend

A005674 a(n) = 2^(n-1) + 2^[ n/2 ] + 2^[ (n-1)/2 ] - F(n+3).

0, 0, 0, 0, 1, 3, 10, 25, 63, 144, 327, 711, 1534, 3237, 6787, 14056, 28971, 59283, 120894, 245457, 497167, 1004256, 2025199, 4077007, 8198334, 16467597, 33052491, 66293208

Offset: 0

N. J. A. Sloane

a(n) is the number of compositions of n where mixing of even and odd summands occurs. That is, at least one even summand is bracketed by two odd summands, or vice versa. - Gregory L. Simay, Jul 27 2016

			a(6) = a(2*3) = 2^5 - f(9) + 3*2^2 = 32 - 34 + 12 = 10. The 10 compositions are (1,4,1), (3,2,1), (1,2,3), (2,1,2,1), (1,2,1,2), (2,1,1,2), (1,2,2,1), (1,2,1,1,1), (1,1,2,1,1), (1,1,1,2,1).

R. K. Guy, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

R. K. Guy, Letter to N. J. A. Sloane, 1987
Simon Plouffe, Approximations de séries génératrices et quelques conjectures, Dissertation, Université du Québec à Montréal, 1992; arXiv:0911.4975 [math.NT], 2009.
Simon Plouffe, 1031 Generating Functions, Appendix to Thesis, Montreal, 1992

Cf. A079289, A027558 divided by 2.

A005674:=-z**4/(2*z-1)/(z**2+z-1)/(-1+2*z**2); # [Conjectured by Simon Plouffe in his 1992 dissertation.]

From Gregory L. Simay, Jul 27 2016: (Start)
If n=2k, then a(n) = 2^(n-1) - 2*A079289(n) + 2^(n/2 - 1) + F(n).
If n=2k-1, then a(n) = 2^(n-1) - 2*A079289(n) + F(n). (End)

A193923 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=(x+1)^n and q(n,x)=Sum_{k=0..n}F(k+1)*x^(n-k), where F=A000045 (Fibonacci numbers).

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 1, 3, 5, 8, 1, 4, 8, 13, 21, 1, 5, 12, 21, 34, 55, 1, 6, 17, 33, 55, 89, 144, 1, 7, 23, 50, 88, 144, 233, 377, 1, 8, 30, 73, 138, 232, 377, 610, 987, 1, 9, 38, 103, 211, 370, 609, 987, 1597, 2584, 1, 10, 47, 141, 314, 581, 979, 1596, 2584, 4181, 6765

Offset: 0

Clark Kimberling, Aug 09 2011

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

The row sums equal A079289(2*n). - Johannes W. Meijer, Aug 12 2013

			First six rows:
1
1...1
1...2...3
1...3...5....8
1...4...8....13...21
1...5...12...21...34...55

Michel Marcus, Rows n=0..100 of triangle, flattened
T. G. Lavers, Fibonacci numbers, ordered partitions, and transformations of a finite set, Australasian Journal of Combinatorics, Volume 10(1994), pp. 147-151. See triangle p. 151 (with rows reversed and initial term 0).

Cf. A193722, A193924.

Cf. A001906 (Fibonacci(2*n)).

T := proc(n, k) option remember: if k = 0 then return(1) fi: if k = n then return(combinat[fibonacci](2*n)) fi: T(n, k) := T(n-1, k-1) + T(n-1, k) end: seq(seq(T(n, k), k=0..n), n=0..9); # Johannes W. Meijer, Aug 12 2013

p[n_, x_] := (x + 1)^n;
q[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193923 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193924 *)

T(n, k) = Sum_{p=0..k} binomial(n+k-p-1, p). - Johannes W. Meijer, Aug 12 2013

T(n, n) = Fibonacci(2*n) for n>=1. - Michel Marcus, Nov 03 2020

cp's OEIS Frontend

A005674 a(n) = 2^(n-1) + 2^[ n/2 ] + 2^[ (n-1)/2 ] - F(n+3).

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